- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤ (values match St000094The depth of an ordered tree.)

Values

[1] => [1] => [1,0] => [[]] => 1

[1,2] => [1,2] => [1,0,1,0] => [[],[]] => 1

[2,1] => [2,1] => [1,1,0,0] => [[[]]] => 2

[1,2,3] => [1,2,3] => [1,0,1,0,1,0] => [[],[],[]] => 1

[1,3,2] => [1,3,2] => [1,0,1,1,0,0] => [[],[[]]] => 2

[2,1,3] => [2,1,3] => [1,1,0,0,1,0] => [[[]],[]] => 2

[2,3,1] => [3,2,1] => [1,1,1,0,0,0] => [[[[]]]] => 3

[3,1,2] => [3,2,1] => [1,1,1,0,0,0] => [[[[]]]] => 3

[3,2,1] => [3,2,1] => [1,1,1,0,0,0] => [[[[]]]] => 3

[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => 1

[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => 2

[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [[],[[]],[]] => 2

[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => 3

[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => 3

[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => 3

[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [[[]],[],[]] => 2

[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [[[]],[[]]] => 2

[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => 3

[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => [[[[],[]]]] => 3

[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => 3

[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => 3

[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4

[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => 1

[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => 2

[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => 2

[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 3

[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 3

[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 3

[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => 2

[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => 2

[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 3

[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => 3

[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 3

[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 3

[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 4

[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => 2

[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => 2

[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => 2

[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 3

[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 3

[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 3

[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => 3

[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 3

[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 4

[2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => 4

[2,3,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[2,4,1,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [[[[],[]]],[]] => 3

[2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => 4

[2,4,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 4

[2,4,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => 4

[2,4,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => 4

[2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => 4

[2,5,3,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => 4

[2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => 4

[2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => 3

[3,1,2,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 3

[3,1,4,2,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 4

[3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => 4

[3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => 3

[3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 3

[3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 4

[3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => 4

[3,2,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 4

[3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 4

[3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

[3,5,1,2,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => 4

[3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5

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Description

The depth minus 1 of an ordered tree.
The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to St000013The height of a Dyck path..

Map

Demazure product with inverse

Description

This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.

Map

to ordered tree

Description

Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path $D$ of semilength $n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths $D_1,\dots,D_k$ of respective semilengths $n_1,\dots,n_k$ (so one has $n = n_1 + \dots n_k$) each of which has no returns.
Denote by $\tilde D_i$ the path of semilength $n_i-1$ obtained from $D_i$ by removing the initial up- and the final down-step.
This map then sends $D$ to the tree $T$ having a root note with ordered children $T_1,\dots,T_k$ which are again ordered trees computed from $D_1,\dots,D_k$ respectively.
The unique path of semilength $1$ is sent to the tree consisting of a single node.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.